In this chapter, I apply the validated static bending test in order to mechanically characterize low aspect ratio, rigid ZnO NRs. In contrast with the references [21, 46, 51, 59, 62-63, 65-70], I performed my measurements on highly ordered ZnO NRs perpendicularly standing on their substrate. Up to my knowledge this thesis (and the corresponding own publication [T4]) is the first work on the mechanical characterization of wet chemically grown vertical ZnO NRs.
6.1. Experimental subsection
The NRs arranged in a triangular lattice were synthesized by selective area wet chemical growth method onto the surface of a c-axis oriented (Zn-polar), hydrothermal ZnO single crystal (CrysTec GmbH) (see subsection 4.1.). In order to make the NRs accessible for the AFM cantilever the ZnO wafer was cleaved across the patterned area (Fig. 38d). Hence, I obtained perpendicularly standing NRs at the edge of the substrate.
The bending experiments were carried out in SEM by a soft silicon AFM tip (Fig. 38a-b) (MikroMasch CSC17/Cr-Au), which was mounted on a nanomanipulator arm (see subsection 5.1.). Prior to testing, the resonance frequency of the cantilever (f) was measured by our scanning probe microscope. Contrary to the case of silicon-nitride cantilevers, here the calibration is much easier, since silicon cantilevers are manufactured by wet etching from the single crystalline wafer and therefore their crystallographic orientation and hence Young‟s modulus is well known. One only has to determine the dimensions (by SEM) and the resonance frequency (by AFM) of the cantilever, and the spring constant can be analytically calculated with high confidence. In order to ensure a permanent contact between the probe and the NR during the bending test and to avoid slipping from the edges, a U-shaped incision was etched by focused ion beam at the end of the tip (Fig. 38c). The coarse positioning of the AFM tip toward the sample was done with the robotic arm, while the fine positioning as well as the manipulation of the NRs was achieved by moving the sample with the stage. The schematic of the bending experiment is shown in Fig. 31a-b. In my systematic characterization carried out on six individual NRs, the bending load was applied in the
<11 0> crystallographic direction, i.e. perpendicular to the vertical axis of the NR and parallel to the long symmetry axis of the hexagonal cross-section. In order to investigate the
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effect of the load direction, the bending experiment, in one case, was also done in the <10 0>
direction, i.e. perpendicular to the long symmetry axis of the hexagon (Fig. 39c inset). As in the case of InAs NWs in subsection 5.1., stress was applied via moving the stage and hence the NR towards the tip and thus deflecting the loading cantilever. Simultaneously, two snapshots were recorded at each bending test, one in stressed and the other one in relaxed state. By overlapping the two snapshots both NR and tip deflections (y and Y) can be determined by image analysis.
Figure 38. SEM micrographs of the MikroMasch CSC17/Cr-Au AFM cantilever in lower magnification (a) and in higher magnification (b) showing the pyramidal tip at the end. A U-shaped incision etched by FIB at the edge of the tip (c) to avoid slipping of the NR during manipulation. Lower magnification SEM image of the cantilever and tip touching the array of vertical ZnO NRs at the edge of the cleaved wafer (d).
6.2. Results and discussion
According to the SEM observation the ZnO NRs are arranged in the desired lattice (Fig. 39a).
Each NR is c-axis aligned and consists of two parts: a shaft at the bottom, which was formed by filling the hole in the PMMA layer and a longer upper part having hexagonal cross-section (Fig. 39b), which was formed „„naturally‟‟ during the chemical growth. The hexagonal cross-sections are collectively aligned according to the crystal orientation of the substrate. The length of the NRs varied in the range of 1.3–1.5 µm, while the Feret‟s diameter of the (0001) hexagonal top face is 105 ± 8 nm.
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Figure 39. Top and perspective view scanning electron micrographs of the epitaxially grown NR array (a and b).
Randomly selected NRs standing at the cleaved edge were bent by a calibrated AFM tip (c) in the <11 0>
(arrow marked by F1) and <10 0> (arrow marked by F2) crystallographic directions (c inset). Side-view image taken on the NRs (d) shows the non-uniform cross-section along the vertical axis. During the calculation of the BM the NR geometry is divided into a lower truncated circular and in an upper truncated hexagonal cone (TRCC and TRHC) (e).
During the NR bending test I found that even strongly deflected NRs returned back to their original position when released. Therefore all deflections measured in this work can be assumed to be purely elastic. In a certain static bent state (Fig. 39c) the two spring forces of AFM cantilever and of ZnO NR are in equilibrium; therefore, it can be calculated by multiplying the AFM cantilever deflection (Y) and its spring constant (kSi)
. (21)
Although kSi can theoretically be calculated purely from the geometrical parameters and from the Young‟s modulus of Si (ESi) I followed instead the method suggested by Cleveland et al.
[133] to eliminate the purely measurable cantilever thickness value
, (22)
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where w, f0, L, and ρSi are the width, unloaded resonant frequency, length, and mass density of the rectangular Si cantilever, respectively. The geometrical parameters of the cantilever were estimated from SEM images (Fig. 38a). The obtained kSi value is 9.33·10-2 N/m, while the bending force value in each experiment is shown in Table 5.
Table 5. Summary of the bending experiments. The columns from left to the right denote: NR number (NR), load (F), lower (d1) and upper (d2) diameter of the truncated circular cone, lower (d3) and upper projected width (d4) of the considered hexagonal cone, height of the truncated circular cone (l1), point of the applied force (l), NR deflection (NR defl.) and calculated BM (EBM) using equation (28).
The BM of a perpendicularly standing beam fixed at the bottom and pushed by a lateral force is most commonly calculated by solving the static Euler–Bernoulli equation on a uniform cross-section (Eq. 19). Here the second moment of inertia can be calculated by assuming either hexagonal (Eq. 14) or circular cross-section
, (23)
where r is the radius of the circle. However, as shown in Fig. 39d the change of the diameter along the vertical axis is obvious, which can cause a multiplied uncertainty in BM because of
NR
56 158 137 140 128 318 1361 104 20.3
6 61 157 139 143 125 380 1270 54 33.2
Average 32.2 ± 7.4
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the fourth-power dependence (Eqs. 14 and 23). Therefore, in case of as prepared vertical NRs the implementation of a non-uniform beam model is essential. In the specific case of elastic beams Castigliano‟s theorem says that the derivative of the total strain energy (U) of an elastic beam with respect to the load (Fi) is equal to the deflection ( ) corresponding to the load [128]:
.
(24)Assuming that the strain energy due to traverse shear loading is negligible, the strain energy resulting from bending of the NR can be calculated as follows:
,
(25)where the x axis runs along the NW with origin fixed at the height of the applied load, and M=Fx is the internal bending moment, which is a function of x. Therefore the deflection of NR can be calculated as follows:
.
(26)Or, if we fix the origin of the coordinate system at the bottom of the NR we can also write:
.
(27)To describe the practical geometry, the ~300 nm high bottleneck-shaped part at the bottom of the nanocrystals can be approximated as a truncated right circular cone (TRCC), while the upper part as a truncated right hexagonal cone (TRHC) (Fig. 39d–e). Hence, using Eqs. (14), (23), and (27) the BM can be expressed by the following form:
, (28)
where d1, d2, l1, d3, d4 are the lower and upper diameter of the TRCC, the height of TRCC, the
<11 0> projected width ( ) at the bottom and at the top of the considered TRHC, respectively. These parameters are determined from side view SEM images captured after each bending test by setting the electron beam parallel to the direction of the load (e.g. Fig.
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39d). I examined six NRs under different bending states. Detailed geometrical parameters and the obtained BM values are summarized in Table 5.
The obtained bending moduli range from 20 to 44 GPa. The average value of 32.2 ± 7.4 GPa is significantly lower than the Young‟s modulus of bulk ZnO in the [0001] direction (140 GPa) [72]. The strong discrepancy is usually attributed to different effects such as surface stress or electromechanical coupling [63]. To answer this question a further systematic investigation is required. However, the modulus reported in this chapter is similar to the one which was measured by Manoharan et al. [63] (44 GPa) by applying also in situ static bending experiment. For comparison, the BM was calculated in the simpler way as well, i.e.
by assuming a beam with uniform circular cross-section (Eq. 19) instead of the two-part mechanical model. The homogeneous second moment of inertia was estimated by Eq. (23), in which the half of the corresponding d1 diameter (at the bottom of the TRCC) was considered as the radius of the circle in the case of each NR. The so calculated mean BM of 24.8 ± 7.1 GPa is lower than that provided by the more sophisticated model, therefore the changing cross-section along the vertical axis cannot be neglected and the non-uniform beam model is essential.
The comparative bending tests carried out on the same NR along <11 0> and <10 0>
directions (Fig. 39c inset) could not reveal significant load direction dependence. The obtained BM values calculated by Eq. (28) were EBM<11–20> = 29 ± 5 GPa and EBM<10–10> = 34 ± 5 GPa. Note, that the second moment of inertia for hexagonal cross-section (14) is valid for both horizontal and vertical axis through the centroid, therefore Eq. (28) is also valid for both load directions.
In conclusion, I have performed a well controlled and validated static experiment to determine the BM of low aspect ratio homoepitaxially grown vertical ZnO NRs. In the mechanical test the ZnO NRs were deformed along the <11 0> direction by a calibrated AFM tip in an SEM chamber. I proposed a non-uniform beam model for the BM calculation, which takes into the account the changing cross-section of the NR along the vertical axis. I found that the average value of 32.2 ± 7.4 GPa is significantly higher than the calculated one which assumes a homogeneous circular cross-section along the whole length (24.8 ± 7.1 GPa). The demonstrated technique can be later also a powerful tool for the calibration of the proposed integrated nanoforce sensor. Moreover in the future I am also planning to carry out fracture tests by the same manner in order to determine the fracture strength of the NRs.
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